2017-08-01

Jean-Morlet Chair - Research Talks - Lemanczyk/Ferenczi

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source: Centre International de Rencontres Mathématiques    2016年11月2日

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 1 1:27:36
Abstract: * The early results of Ramsey theory:
Hilbert's irreducibility theorem, Dickson-Schur work on Fermat's equation over finite fields, van der Waerden's theorem, Ramsey's theorem and its rediscovery by Erdos and Szekeres.
* Three main principles of Ramsey theory:
First principle: Complete disorder is impossible. Second principle: Behind every 'Partition' result there is a notion of largeness which is responsible for a 'Density' enhancement of this result. Third principle: The sought-after configurations which are always to be found in large sets are abundant.
* Furstenberg's Dynamical approach:
Partition Ramsey theory and topological dynamics Dynamical versions of van der Waerden's theorem, Hindman's theorem and Graham-Rothschild-Spencer's geometric Ramsey. Density Ramsey theory and Furstenberg's correspondence principle Furstenberg's correspondence principle. Ergodic Szemeredi's theorem. Polynomial Szemeredi theorem. Density version of the Hales-Jewett theorem.
* Stone-Cech compactifications and Hindman's theorem:
Topological algebra in Stone-Cech compactifications. Proof of Hind-man's theorem via Poincare recurrence theorem for ultrafilters.
* IP sets and ergodic Ramsey theory:
Applications of IP sets and idempotent ultrafilters to ergodic-theoretical multiple recurrence and to density Ramsey theory. IP-polynomial Szemeredi theorem.
* Open problems and conjectures
If time permits: * The nilpotent connection, * Ergodic Ramsey theory and amenable groups
Recording during the Jean-Morlet Chair doctoral school: "Spectral Theory of Dynamical Systems and Related Topics" the October 17, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)
Filmmaker: Guillaume Hennenfent
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Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 2 1:30:01
Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 3 1:12:02
Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics- part 4 1:01:01
Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 5 1:26:29
Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 6 1:30:33
Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 7 1:13:51
Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 8 58:42

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